Tracing the dynamics of superconducting order via transient terahertz third-harmonic generation

Ultrafast optical control of quantum systems is an emerging field of physics. In particular, the possibility of light-driven superconductivity has attracted much of attention. To identify nonequilibrium superconductivity, it is necessary to measure fingerprints of superconductivity on ultrafast timescales. Recently, nonlinear THz third-harmonic generation (THG) was shown to directly probe the collective degrees of freedoms of the superconducting condensate, including the Higgs mode. Here, we extend this idea to light-driven nonequilibrium states in superconducting La2-xSrxCuO4, establishing an optical pump–THz–THG drive protocol to access the transient superconducting order-parameter quench and recovering on few-picosecond timescales. We show in particular the ability of two-dimensional TH spectroscopy to disentangle the effects of optically excited quasiparticles from the pure order-parameter dynamics, which are unavoidably mixed in the pump-driven linear THz response. Benchmarking the gap dynamics to existing experiments shows the ability of driven THG spectroscopy to overcome these limitations in ordinary pump-probe protocols.

Supplementary Text S1: Two methods for measuring the TH dynamics under the photoexcitation The Third harmonic (TH) field driven by high-field multi-cycle THz pulse is a preferential spectroscopic tool to investigate the superconducting order.By applying a combination of an optical pump pulse and a multicycle TH pulse, we investigate non-equilibrium superconductivity.Unlike the conventional time-domain THz probe spectroscopy, which represents the non-equilibrium linear electronic properties, the TH spectroscopy allows us to disentangle the nonlinear properties of the system from the pump-induced effects on the linear response, as due to quasiparticle excitations.We termed our protocol an optical pump -TH drive experiment, as opposed to a usual pump-probe scheme.
Tilted pulse front technique generates single-cycle broadband THz field.We filtered it to obtain a narrow-band driving field with the frequency centered at ω=0.7 THz and average duration about 10 ps by using band-pass filter.To measure the driven TH electric field change under the photoexcitation, we focused the optical pump beam with the pulse duration of 100fs onto the sample with a beam diameter 600 µm comparable to the focused THz pulse 560 µm.The data acquisition processes were in two ways, 1) Scanning pump delay (1D scan) and 2) Scanning pump and THz delay lines simultaneously (2D scan).
Figure S1 (a) depicts the 1D scan.Here, tTHz and tg are kept fixed to record the change of the TH field while sweeping ∆t.For this measurement it is necessary to fully suppress the leaked fundamental driving field (FH) field.We achieve this by installing two bandpass filters (2.1 THz) after the sample.The optical pump beam is modulated by a chopper obtaining the relative changes of the TH field.
Figure S1 (b) shows the 2D scan scheme to collect the full THz spectrum.The static transmitted THz field is scanned by moving the tTHz with the fixed gate time delay tg.To collect the full THz spectrum including both FH and TH components, we used the one bandpass filter (2.1THz) after the sample.While the scanning, the optical pump pulse tpump moves with the fixed pump-drive time delay ∆t simultaneously.Each scans are operating with systematically varying ∆t.In this procedure, we obtained the nonlinear sample response between the THz field and the optical pump pulse.

Supplementary Text S2: Interference effect between optical pump and TH drive field
Figure S2 describes the way of extracting the photoexcited interference feature in a superconductor.At the early pump-drive time delay Δt a prominent strong oscillatory signal emerges on top of the background response, as seen in Fig. S2 (a,b).To extract the oscillatory signal, we fit the nonoscillatory background signal in analogy to previous studies on other correlated materials (53-55).The fit function is designed as where ∆t is the time delay,   is the time constant for the excitation of photo-carriers,  1 and  1 are the time constant of the exponential decay, and A1, A2, c, ∆t0 are constants.Subtracting the fits results in Fig. S2 (c,d).As explained at the end of the previous section, in the 1D pump scan we select only the   = 3Ω dynamics of   by adding an additional band-pass filter.As a consequence the frequency of the Δt oscillations are   = ±2 (1.4 THz) and   = ±4 (2.8 THz), see Fig. S2 (e,f).Apart from the (non-oscillatory) contribution of the static THG signals, Fig. S2 (c,d) are equivalent to the projection of the 2D scans with digitally filtered to obtain only TH component as described in the Supplementary Text S3.

Supplementary Text S3: Theoretical calculation of pump-TH drive dynamics
To pinpoint the origin of THG modulations with even multiples of  as reported in the main text, we recorded spectra of the transmitted signal as a function of gate time   and pump-drive delay .The resulting spectrum is shown in Fig. S3 (a).Fig. S3 (b-c) show the data after a Fourier transform first along the   axis, and then along the -axis, with corresponding frequency variables   and   , respectively.In what follows, we will first focus on the discussion of the experimental data in (  ,  )-space shown in Fig. S3 (c).The experimentally recorded transmitted field is directly related to the induced current () =  (1) () +  (3) () which we further decompose into first-and third-order contributions of the applied electromagnetic vector potential  =  THz +  pump .Signatures of the fifth-order current  (5) () are also observed, as detailed in the below.
The experimental setup consists of three pulses.The optical pump  pump , the THz drive  THz , and a gate pulse at which the parallel electric field component of the transmitted beam is recorded.Here we define the time origin  =   = 0 to be the center of the THz drive.We denote functional forms of the pulse shapes centered around the origin by a bar, so that  pump () =  ҧ pump ( + ).In frequency space the pump delay  hence results in a phase shift  pump () =  ҧ pump () − .The linear response current is given by  (1) (  ) = (  ) (1) (  )(  ) ≈  (1) (  ) THz (  ) , (S2) where  (1) is the linear susceptibility and () is the transmission function of a 3-bandpass filter shown in Fig. S4.Only the probe pulse generates a response in the THz regime.Since  THz is unaffected by  and is a narrowband multi-cycle pulse centered at frequency , the linear response yields the signal at (  ,   ) = (±, 0) in Fig. S3 (c).
For the optical pump pulse used in the experiment,   ⊗   has an approximately Gaussian peak centered around zero with FWHM of ~5 THz, consistent with the broad diagonal feature observed between |  | ⪅ 3.The absence of signal at low frequencies (between |  | ⪅ ) in Fig. S3 (f) is a consequence of the 3 bandpass filter used in the experiment.We can intuitively understand Eq. (S4) within the diagrammatic representation shown in Fig. S5.Here, one photon of the THz drive pulse (black wavy line) combines with two photons of the pump pulse (blue wavy lines) to produce the nonlinear current (red wavy line).The full grey bubble represents the nonlinear susceptibility  (3) that embodies the microscopic details of the material.In the case of both the quasiparticle and the Higgs mode contributions, such renormalization would lead to a peak in the susceptibility at the characteristic resonance energy of 2.To conserve energy, the frequency of the three photons must match the frequency of the induced current.The probe photon can only contribute with energies ±.While the photon energies of the pump pulse lie in the optical regime, i.e.  ∼ ±eV, their frequency spectrum is rather broad and two such high energy photons can combine in a difference frequency (DF) process to yield the total energy  ∓  in the THz regime.Due to the pump delay , both pump photons pick up a phase that results in phase oscillations in  manifest in the diagonal spectral features at  Δ =   ±  in Fig. S3 (c) (marked by the green rectangles boxes in Fig. 2 (c) in the main text).Following the same steps detailed above, the diagonal stripes at  Δ =   ± 3 in Fig. S3 (c) (marked by the red rectangle box in Fig. 2 (c) in the main text) can be related instead to the fifth-order current  (5) .
The 2D experiment described above differs from the 1D experiment described in the main text in two ways.First, the 1D experiment was performed at two selected values of the gate time   =  ± , corresponding to the maximum(minimum) field strength of the THz probe pulse.Second, the 2D experiment was performed using a single 3 bandpass filter of transmission (), see Eq. (S2), while two such filters, () 2 , were used in the 1D experiment.The experimentally recorded transmission function of the filter is shown in Fig. S4.To verify the consistency of the two experiments, we bridge these difference in two steps.We first digitally apply the second filter to the 2D data according to  filtered (3) (  ) = (  ) (3) (  ) (S5) The filtered data in the (  ,   )-domain are shown in Fig. S3 (f).We see that the remaining signal is concentrated in a narrow band around   = ±3.Fourier transforming the axes back into time domain yields Fig. S3 (d-e).
Next, we tune the gate time   to the maximum (minimum)  ± of the signal as indicated by blue (orange) lines in Fig. S3 (d).In principle  ± should be independent of , but we allow for a small -dependence to compensate for small drifts of the pulse position when  is experimentally scanned.
A plot of the signal along the blue (orange) lines is shown in Fig. S6 (a).We observe oscillations in  in agreement to the 1D data of Fig. S2 (a,b) (Fig. 3 (a,b) in the main text).The corresponding Fourier transform in Fig. S6 (b) reveals the frequency of these oscillations to be 2 and 4.
From the preceding discussion, we can understand the origin of the 2 and 4 oscillations.The measurement for a specific gate time  ± corresponds to an integration over a range of   frequencies in Fourier space.This is approximately equivalent to a projection of the spectrum in Fig. S3 (c) on the   -axis.However, the presence of the filter removes all but the   = ±3 signal.Hence, mostly the signal at   = ±3 contributes.Therefore, the 1D measurement effectively probes the points (  ,   ) = (±3, 0), (±3, ±2), and (±3, ±4) in the 2D FT spectrum shown in Fig. S3 (f).The same effect is easily seen in time domain in the 1D differential scans reported in Fig. 1 (c) and Fig. 3 (a,b) of the main text.In this case the THz field-change   scales as the amplitude of the pump-induced contribution  (3) (  , Δ) to the non-linear current given by Eq. (S4).It is then expected that it oscillates as a function of  at 2 and 4, the latter being suppressed by the filtering procedure as shown by comparing Fig. S3 (c) and (f).

Supplementary Text S4: The light-induced changes of TH at long time delays
Figure S7 shows the light-induced changes in TH at a pump-probe delay of ∆t = 60 ps.According to Eq. ( 1), one would expect the signal to scale with   2 .i.e. with the pump fluence.Such a behavior is indeed observed for the signal at long time delays, where the TH amplitude changes show a linear dependence on optical pump flunence F (∝   2) as it increases.Additionally, the absence of saturation behavior suggests that we may not have reached the saturation regime, as clearly demonstrated at pump fluences exceeding 50 μJ/cm 2 [14].

Supplementary Text S5: The non-monotonic behavior of the maximum peak-field changes of TH as a function of the pump fluence
The maximum peak-field changes of the TH signal, δE TH MAX of Fig. 3C, exhibit a non-monotonic behavior as a function of fluence F. As the pump fluence increases, a non-linear increase feature is evident, resembling observations in photoexcited optical and linear THz responses, suggesting a saturation behavior indicative of the destruction of superconductivity [14].Also, a comparable depletion feature above a critical pump fluence has been found in photo-excited excitonic insulators [42, 43].In analogy to these previous studies, the fit function is designed as

Supplementary Text S6: Comparison the pump induced TH changes to unpumped TH amplitude
Figure S8 shows the direct comparison of the TH field changes under the photoexcitation to the unpumped TH field response.The onset of the TH field changes signal below 30K appears at temperatures where the unpumped TH field intensity peaks.As discussed in ref. 25 this suppression of the static TH signal is due to strong screening effects that take place in the superconducting phase.This finding suggests that the sign-change of the pump-induced variation of the TH signal can be explained by a predominant effect of the local-field screening.

Supplementary Text S7: 2D pump-drive spectra along sheared time axis τ
To examine the features of the transient dynamics of the linear FH component and the nonlinear TH signal distinctly, we now consider a different scheme: namely, we introduce a new variable  =   + ∆ and plot the 2D spectra as a function of (  , ) instead of (  , ∆), as shown in Fig. S9 (b).Geometrically, this corresponds to a 45° shear of the original 2D spectrum in Fig. 2A.This physically corresponds to collecting individually traces of the experimental data at fixed pump-gate delay, or adopting the method 2 described in Ref. 46.This is equivalent to the experimental scheme shown in Fig. S9 (a).By fixing the time between the pump pulse and gate pulse and sweeping both pulses while the THz pulse is fixed, every point of the THz waveform is measured with same pumpprobe delay time .
As shown in Fig. S9 (c) after the Fourier transform along the gate time tg, the FH now shows coherent modulations as a function of pump-drive delay time τ (Plots of the FH and TH intensity changes are shown in Fig. S9 (e)).The spectral feature of background excitation-relaxation dynamics are marked by cigar-shaped contours (blue: FH, violet: TH).These features are tilted by 45° compared to the corresponding in the (  ,   )-representation of (Fig. 2C).In a similar vein, the horizontal feature at the FH component shown in frequency with broadened from ω to 2ω marked by green rectangle in Fig. S9 (d).In the tiled 2D spectra shows c The FH modulations indicates that the FH modulations is highly related on the TH modulations.
( −   ) + .(S6) The first term corresponds to the saturating part, where F denotes the optical pump fluence, and Fc characterizes the position of the critical fluence.The second term in Eq. (S6) describes the linear decrease behavior, where w representing the width.The factors C1, C2 describe the amplitudes of the two components, and d is a constant.

Fig. S1 .Fig. S2 .
Fig. S1.Experimental scheme for the optical pump-TH drive experiment.The three time points tpump, tTHz and tg are shown.t=tTHz=0 defines the center of the THz envelope and it is set conventionally to zero.The center t=-tpump <0 of the pump pulse is located at negative times, and we denote the pump probe delays as ∆t= tTHz -tpump.In this convention, ∆t is always positive when the pump pulse precedes the THz probe.(a) 1D pump scheme.tTHz and tgate are fixed to record the change of the TH field while sweeping ∆t.To suppress the dominant FH field, two band pass filter (BPF) are installed after the sample.Here, ∆t changes in one scan.(b) 2D time-domain scans scheme.The tpump and tTHz are simultaneously varying while the tg =0 is fixed.The ∆t is fixed during a scan in tgate, subsequently each scan has a different value for the time distance ∆t.
Fig. S3.2D spectra in time and frequency domain.(a-c) Experimentally recorded data for 2D scans in (a) time domain, (b) time-frequency domain and (c) frequency domain.The experiment was performed using a single 2.1 THz band pass filter, so that leakage of FH can be observed.(df) Projection of the 2D scan results on the experimental conditions used to acquire the 1D scans.Since in 1D scans two band-pass 2.1 THz filters have been used, a second 2.1 THz band pass filter has been digitally applied to the data of panel (a-c) to match the experimental setup used to produce the data shown in Fig. 1(c) of the main manuscript and in Supplementary Text S2.This completely filters out the FH signal, as seen in panels (e) and (f).In addition, the second filter also removes the TH modulations at   = 4 that are still visible in panel (f).The data Fourier transformed back to real time are shown in panels (d) and (e).

Fig. S4 .
Fig. S4.Transmission function () of the band pass filter.Blue curve shows the experimentally recorded data, orange curve shows a fit used in the data processing routine.

Fig. S5 .
Fig. S5.Diagrammatic representation of the non-linear response.Diagrammatic representation of all the contributions to the third-order nonlinear current (3) () (red wiggly line).The thirdorder susceptibility (3) , including in principle all diamagnetic, paramagnetic and mixed terms in the presence of disorder (see Ref. 56-58 for more details), is schematically represented here as a full grey bubble.(a) Process involving two THz and one optical pump photon: the nonlinear current response in this case is in the optical range as the pump field.(b) FH and TH generation process of the optical pump field.(c) FH and TH processes of the THz field, which provides the contributions to the  and 3 signals measured in the experiments.(d) Process involving two pump photons (blue) and one THz photon (black): in order to obtain a THz response a difference-frequency process involving the two pump photons needs to be considered.

Fig. S6 .
Fig. S6.Analyzing dynamic patterns of the digitally filtered data in Fig. S3 (d).(a) Plot of the maximum (blue) and minimum (orange) of the digitally filtered data of Fig. S3 (d) as a function of .The data correspond to the slice marked by horizontal lines in Fig. S3 (d).The Fourier transform of the data respect to  in (b) reveals 2 and 4 components of the modulations in .

Fig. S7 .
Fig. S7.Fluence dependence of the TH peak field changes at long time delay.Optical pump fluence dependence of the extracted TH peak-field changes at the pump-probe delay time ∆t=60ps.

Fig. S9 .
Fig. S9.Intrinsic response of optical pump-TH drive signal.(a) Equivalent experimental scheme to shearing 2D spectrum.The tpump and tgate are simultaneously varying while the tTHz =0 is fixed.The time difference between the pump and the gate, τ, is fixed during a scan, subsequently each scan has a different value for the time distance  =   + ∆.(b) The sheared data in x-axis (∆t) of Figure 2(a).The spectrum is normalized by the maximum value of the transient THz field.(c) Fourier transform along the tg axis of (a).The spectrum is normalized by the equilibrium value of the FH magnitude.(d) A 2D Fourier transform of (b) along the sampling tg and pump-drive delay axis τ.Cigar-and rectangular-shaped features correspond to pumped dynamics and emerged photoexcited harmonics, respectively.(e) Transient dynamics of the FH and TH intensity (IFH, ITH) extracted from (c).The value is normalized by the dataset of the static case before the pump pulse arrives.